Visual Calculus - Riemann Sums

Problem:
Graphically illustrate the definition of Riemann Sums for the function, y = f(x) with domain [a, b], whose graph is

The blue line in the graph is part of the x-axis.

Note. The definition of Riemann Sums will be given in the development that follows.

Visualization:

First, we pick some positive integer n. For our illustration, we shall pick n = 10.
We now subdivide the interval interval into n equal subintervals.
Each of the new subintervals has length
We will label the endpoints of the new subintervals:

a₀, a₁, a₂, ..., a₁₀
which is called a partition of [a,b].
In each of the subintervals [a_i-1, a_i], we pick a number x_i and draw a line segment perpendicular to the x-axis from the point (x_i,0) to a point on the graph of the function, (x_i, f(x_i)).
As in this animation, we then construct rectangles which have the line segments as their height and the subintervals as their base.
If each f(x_i) > 0 then the area of the i^th rectangle is

and the sum of the areas of the rectangles is then:

More generally, we do not require that f(x_i) > 0 as we define

A Riemann Sum of f over [a, b] is the sum

If you want to view some additional graphs illustrating Riemann Sums with different values of n and different choices of x_i's, then make your choices from the following two groups of options:

Note that the Riemann sum when each x_i is the right-hand endpoint of the subinterval [a_i-1, a_i] is

when each x_i is the left-hand endpoint of the subinterval [a_i-1, a_i] is

and when each x_i is the left-hand midpoint of the subinterval [a_i-1, a_i] is

Summary of the material above.